#7712727

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,7,8,5)(11,14)(13,21)(16,18)(17,20)(28,36)(32,37)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(26,30)(31,33)(32,37)(38,39);(9,11,15,14)(10,16,12,18)(13,22,21,19)(17,24,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12,15,10)(11,16,14,18)(13,20,21,17)(19,23,22,24);(9,13,15,21)(10,17,12,20)(11,19,14,22)(16,23,18,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,14,15,11)(10,16,12,18)(13,22,21,19)(17,23,20,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619923);